MCM 2019 Problem C Solution

We participated in the MCM 2019 and chose Problem C. This is our solution.

Also, we made some beautiful gifs to visualize the spread of opioids. You can download them and their code in this repo.

Acknowledgement

The latitude and longitude data of 461 counties come from SimpleMaps.com, which is for free. Great Thanks!

Thanks, my teammates, without whom I have no way to finish this.

Examples of spread map gifs

Notice that it is not our model predictions; it's just a visualization of the given historical data.

All drugs

(which I mean TotalDrugReportsCounty in the problem file MCM_NFLIS_Data.xlsx)

Heroin

Fentanyl

Explanation of our model

Overall

We use one state-vector to describe the drug use state of one year, and a transfer-matrix to make the transfer of one year to the next year. And when you want to predict before just use an inverse-transfer-matrix, which is very convenient. So our model would be like this:

or as the figure below illustrated (461 is the number of counties).

Of course, the transfer-matrix is not just a matrix; there are more details.

Details of first model

transfer-matrix of our first model, which used to solve the first part, consists of two parts: distance-matrix and correction-matrix. You might find out that our correction-matrix is actually a fully connected network of 461 inputs and 461 outputs, which is trainable, while our distance-matrix was designed by hand. The figure below illustrates this.

distance-matrix is designed to represent to distance of two counties. For example, if two counties are close to each other, we think that one county's drugs is more likely to spread to anthor, which is a natural thinking. And about how exactly we design this distance-matrix, the answer would be: just try.

correction-matrix is designed to correct the distance-matrix and represents some information we don't consider. But actually, if you say: distance-matrix is designed to accelerate the learning of network, I will also agree with that.

Details of second model

Based on the first model, we add a new part, called economic-matrix, which represents the influence of socio-economic parameters, and mainly to one county itself. So the economic-matrix is a diagonal matrix, and its value is determined by one county's socio-economic data. However, how to get each parameter's weight? You don't know each parameter's effect. The solution is another network. And to explain how each parameter influences the economic-matrix, we choose a fully connected network again, because it's linear and the weights of the network can represent the weights of socio-economic parameters. So the structure would be like this:

A shared layer has been used to ensure every socio-economic network has the same weights.

Part of results

loss of the first model

loss of the second model

prediction of the first model

Summary of results

This first model performs not bad, as you can see. But not on every sample. And the second model is not that good, the loss is very high, though decreased a lot. That might because we made some mistakes when pre-processing the socio-economic data. We thought the tags (like 'HC01_VC03') are the same in each socio-economic data file. However, we are wrong. When we found this mistake, it's very close to the deadline, so we left it. But we do believe that the second model would be a good model if fed well. Also, when we try to inverse the model to predict before, we failed again, some bad results come out, and it's not solved now.

Small details about our model

We don't use any activation function, because we want to make an inverse to predict before. Though we know it will be more reasonable if we put a relu before the final output, because the number of drug use is all positive.

Why we don't use more complex network like CNN, or even LSTM? It's also for making convenient inverse.

Usage

solution folder includes three code files:

plot_gif.py generate spread maps in gif format
model1.py the first model
model2.py the second model